$L_2$-cohomology, derivations and quantum Markov semi-groups on $q$-Gaussian algebras

TL;DR

This paper investigates the cohomological properties of quantum Markov semi-groups on q-Gaussian algebras, establishing new conditions under which these algebras satisfy the Akemann-Ostrand property, expanding known parameter ranges.

Contribution

It introduces higher order Hochschild cocycles via gradient forms, analyzes their values in bimodules, and applies these results to extend the range of q for which q-Gaussian algebras satisfy AO+.

Findings

q-Gaussian algebras satisfy AO+ for |q| ≤ dim(H)^{-1/2}

New q-range in low dimensions compared to previous results

Derivations imply the Akemann-Ostrand property under natural conditions

Abstract

We study (quasi-)cohomological properties through an analysis of quantum Markov semi-groups. We construct higher order Hochschild cocycles using gradient forms associated with a quantum Markov semi-group. By using Schatten-$\mathcal{S}_p$ estimates we analyze when these cocycles take values in the coarse bimodule. For the 1-cocycles (the derivations) we show that under natural conditions they imply the Akemann-Ostrand property (using the Riesz transform). We apply this to $q$-Gaussian algebras $\Gamma_q(H)$. As a result $q$-Gaussians satisfy AO$^+$ for $| q | \leqslant \dim(H)^{-1/2}$. This includes a new range of $q$ in low dimensions compared to Shlyakhtenko.